## Guiding of laser modes based on self-pumped four-wave mixing in a semiconductor amplifier

Optics Express, Vol. 13, Issue 9, pp. 3340-3347 (2005)

http://dx.doi.org/10.1364/OPEX.13.003340

Acrobat PDF (574 KB)

### Abstract

We propose that self-pumped degenerate four-wave mixing may be used to produce novel diode laser systems where lasing is based on non-linear guiding of the laser beams inside the active semiconductor. The fundamental process responsible for the interaction is spatial hole burning in semiconductor amplifiers. The gain and index gratings created by the modulation of the carrier density in the active gain medium lead to selective amplification of one spatial mode and suppression of all other modes. This mechanism allows the laser system to be operated far above its threshold with an almost diffraction limited output beam. The third order nonlinear susceptibility of the non-linear material, which determines the strength of the induced gratings, depends on the angle between the interacting beams in the four-wave mixing configuration. It is shown theoretically that a narrow range of angles exist where the induced gratings are strong and where mode suppression of higher order spatial modes are obtained simultaneously. Experimental evidence sustaining these findings is given.

© 2005 Optical Society of America

## 1. Introduction

^{3}–10

^{4}have been obtained [1

1. H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. **QE-22**, 1349–1354 (1986). [CrossRef]

2. M. Lucente, J. G. Fujimoto, and G. M. Carter, “Spatial and frequency dependence of four-wave mixing in broad-area diode lasers,” Appl. Phys. Lett. **53**, 1897–1899 (1988). [CrossRef]

3. P. Kürz, R. Nagar, and T. Mukai, “Highly efficient phase conjugation using spatially nondegenerate four-wave mixing in a broad-area laser diode”, Appl. Phys. Lett. **68**, 1180–1182 (1996). [CrossRef]

4. R. J. Lang, K. Dzurko, A. A. Hardy, S. Demard, A. Schoenfelder, and D. Welch, “Theory of grating-confined broad-area lasers,” IEEE J. Quantum Electron. **QE-34**, 2196–2210 (1998). [CrossRef]

5. K. Paschke, R. Günter, J. Fricke, F. Bugge, G. Erbert, and G. Tränkle, “High power and high spectral brightness in 1060 nm α-DFB lasers with long resonators,” Electron. Lett. **39**, 369–370 (2003). [CrossRef]

## 2. Laser action based on four-wave mixing in semiconductor amplifiers

*A*

_{3}is generated from a reflection at the external mirror M and this beam diffracts in the four-wave mixing gratings into

*A*

_{2}. From Ref. [3

3. P. Kürz, R. Nagar, and T. Mukai, “Highly efficient phase conjugation using spatially nondegenerate four-wave mixing in a broad-area laser diode”, Appl. Phys. Lett. **68**, 1180–1182 (1996). [CrossRef]

*R*

_{DFWM}=|

*A*

_{4}(

*L*) |

^{2}/|

*A*

_{3}(

*L*) |

^{2}. Using a high-reflectivity coating at one facet of the diode amplifier, a strong beam

*A*

_{1}may be produced from a single internal reflection on the back facet of the laser cavity. The output beam of the laser system

*A*

_{1}(

*L*) is a result of the amplification and four-wave mixing diffraction processes. Due to the angle- and wavelength selectivity of the four-wave mixing gratings, an output beam with high spatial and temporal coherence is produced. Spatial and temporal filters may be added in front of the external, ordinary mirror in order to increase the spatial mode selectivity further.

## 3. Dynamic gratings in semiconductor amplifiers

6. J. Buus and M. Danielsen, “Carrier diffusion and higher-order transversal modes in spectral dynamics of semiconductor-laser,” IEEE J. Quantum Electron. **QE-13**, 669–674 (1977). [CrossRef]

*I*is the injected current,

*q*is the electron charge,

*V*is the active volume,

*N*is the carrier density,

*τ*

_{s}is the spontaneous recombination lifetime,

*D*is the ambipolar diffusion constant,

*E*

_{0}is the total optical field, and, finally,

*g*(

*N*) is the gain that in our analysis is assumed to vary linearly with carrier density, i.e.

*g(N)=a(N-N*where

_{0})*a*and

*N*

_{0}are constants. Temperature variations across the stripe width (coordinate ‘y’ in Fig. 1) that may occur at large injection currents have not been taken into consideration. This effect will only have minor influence on the four-wave mixing process in the case of broad area amplifiers. However, in the case of multiple stripe arrays this effect has to be included [7

7. J.-M. Verdiell and R. Frey, “A broad-area mode-coupling model for multiple-stripe semiconductor lasers,” IEEE J. Quantum Electron. **QE-26**, 270–279 (1990). [CrossRef]

*A*

_{1}and

*A*

_{4}and between

*A*

_{2}and

*A*

_{3}. Due to diffusion of carriers these transmission gratings are much stronger than the reflection gratings in the four-wave mixing geometry [8

8. R. K. Jain and R. C. Lind, “Degenerate four wave mixing in semiconductor doped glasses,” J. Opt. Soc. Am. **73**, 647–653 (1983). [CrossRef]

*k=k*and 〈

_{1y}-k_{4y}=k_{3y}-k_{2y}*N*〉 is the average carrier density. In the following perturbation analysis it is assumed that Δ

*n*<<〈

*N*〉. Inserting Eq. (2) in Eq. (1) we find after some simple calculations that the induced carrier modulation Δ

*n*is given by:

*E*|

_{s}^{2}=(

*ħω*

_{0})/(

*aτ*) is the saturation intensity, and |

_{s}*E*|

_{0}^{2}=|

*E*

_{1}|

^{2}+|

*E*

_{2}|

^{2}+|

*E*

_{3}|

^{2}+|

*E*

_{4}|

^{2}the total intensity, Λ the fringe spacing, and ω

_{0}the optical frequency. In deriving Eqs. (3–4) it is assumed that Δ

*n*<<

*N*

_{0}. The material response is given by the susceptibility [9]:

*g*(

*N*)=

*a*(

*N-N*

_{0}) together with

*N*from Eqs. (2)–(4). The quantity

*β*is the anti-guiding parameter, see e.g. [9], and

*n*is the index of refraction. The amplitude of the spatially varying part

*χ*of the susceptibility responsible for the four-wave mixing process is given by:

_{4WM}*χ*|

_{4WM}^{2}. Since Λ=

*λ*/(2sin(

*θ*/2)), Eq. (6) can be recast:

*χ*is proportional to the optimum value of the non-linear susceptibility when

_{4WM, opt}*θ*=0 and |

*E*

_{0}|

^{2}<<|

*E*|

_{S}^{2}in Eq. (6). Moreover, it should be noted that in deriving Eq. (7) we have assumed that the angle

*θ*between the interacting beams is small. The quantity

*χ*in Eq. (7) determines the strength of the gratings, and in Fig. 2 we have shown

_{4WM}*χ*versus

_{4WM}*θ*calculated from Eq. (7) for different degrees of saturation. The washout of the induced grating due to carrier recombination and diffusion becomes significant as the angle

*θ*increases. The non-linear susceptibility has its largest amplitude for small angles and according to Eq. (7) the amplitude of the spatially varying part of the susceptibility is reduced to half-the-maximum at an angle corresponding to:

*θ*should be less than

*θ*

_{½}if strong non-linear four-wave mixing interaction in-side the active semiconductor is required.

## 4. Mode suppression factor

*δ*between a wave diffracted at the front facet

*z*=0 and at the back facet

*z=L*is given by, see e.g., [10]:

*θ*is the diffraction angle in free space. If this phase difference

*δ*is much larger than unity only the beam incident at the Bragg angle will lead to a diffracted beam and other laser modes will be effectively suppressed. On the other hand, if δ is much smaller than unity all laser modes are diffracted in the grating with almost the same efficiency. As a result,

*δ*plays the role as mode suppression factor. In practice,

*δ*must be somewhat larger than 2 π to have effective suppression of different axial modes in the broad-area amplifier.

*δ=2π*in Eq. (9).

## 5. Output angle condition

*θ*must obey the following condition:

*L*=1 mm,

*n*=3.4,

*D*=13cm

_{a}^{2}/s [6

6. J. Buus and M. Danielsen, “Carrier diffusion and higher-order transversal modes in spectral dynamics of semiconductor-laser,” IEEE J. Quantum Electron. **QE-13**, 669–674 (1977). [CrossRef]

*E*

_{0}|

^{2}=0.5×|

*E*|

_{S}^{2}, and

*τ*

_{s}=1 ns (value for GaAlAs), we obtain

*θ*=4.2° and

_{crit}*θ*

_{½}=8.0° at wavelength λ=810 nm provided the intensity |

*E*

_{0}|

^{2}<< |

*E*|

_{S}^{2}. Accordingly, we conclude that the output angle

*θ*must be larger than 4.2° in order to have good mode suppression and simultaneously it must be less than

*θ*

_{½}=8.0° to have strong gratings in the semiconductor. As the intensity |

*E*

_{0}|

^{2}increases the critical angle

*θ*

_{½}increases and, therefore, it is expected that the optimum angle is moved towards higher values as the output power of the laser increases.

*E*

_{0}|

^{2}is a function of position inside the gain medium. Eq. (11) should, therefore, be considered a first-order approximation. However, by assuming that on average the intensity does not exceed saturation at any position inside the gain medium, we may use Eq. (11) to estimate the angles at which mode suppression and strong four-wave mixing interaction are present at the same time. In Fig. 3, the output angle

*θ*is plotted as a function of the degree of saturation (upper boundary in Eq. (11)) using the same parameters as above. The lower boundary limit is also plotted using the cavity length as parameter (this boundary is independent of the saturation). It is important to note that even though the (averaged) saturation is varied from its minimum to its maximum value, the change in upper limit varies within a factor of √2. Hence, using Eq. (11) in the limit of the low intensity approximation leads in the worst case to a maximum deviation of a factor of √2.

## 6. Experiments

*θ*(unit degrees). The intensity profiles of Fig. 4(a)-Fig. 4(c) have been measured with different pumping levels: In Fig. 4(a), the pump current is

*I*=0.95 A, in Fig. 4(b) the pump current is

*I*=1.23 A, and in Fig. 4(c) the pump current is

*I*=1.40 A. Fig. 4(a)-Fig. 4(c) show a dominant peak and diffraction patterns at both sides of the peak due to diffraction in the induced four-wave mixing grating. In Fig. 4(c) the output power is emitted around an angle

*θ*=6.9° and the output power is 620 mW. The full-width-half-maximum of the central peak in Fig. 4(c) is 0.61°, which is close to the diffraction limit. Fig. 4(d) shows the measured intensity profile of the above configuration where the light path between the external mirror M and the laser diode was blocked. The pump current in Fig. 4(d) was

*I*=1.40 A. Thus, Fig. 4(d) clearly shows that no signal is observed when the mirror M is blocked, i.e. when the induction of gain and refractive index gratings in the diode amplifier is prevented. The emitted angle

*θ*=6.9° in Fig. 4(c) is in good agreement with the theoretical prediction of the optimum output angle condition between 4.2° and 8.0° found from Eq. (11). Furthermore, in Fig. 4(b)-Fig. 4(c) it is observed that the optimum angle is shifted towards higher angles as the output is increased. This observation is also in qualitatively agreement with the theoretical predictions in Section 5.

## 7. Conclusion

## References and links

1. | H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. |

2. | M. Lucente, J. G. Fujimoto, and G. M. Carter, “Spatial and frequency dependence of four-wave mixing in broad-area diode lasers,” Appl. Phys. Lett. |

3. | P. Kürz, R. Nagar, and T. Mukai, “Highly efficient phase conjugation using spatially nondegenerate four-wave mixing in a broad-area laser diode”, Appl. Phys. Lett. |

4. | R. J. Lang, K. Dzurko, A. A. Hardy, S. Demard, A. Schoenfelder, and D. Welch, “Theory of grating-confined broad-area lasers,” IEEE J. Quantum Electron. |

5. | K. Paschke, R. Günter, J. Fricke, F. Bugge, G. Erbert, and G. Tränkle, “High power and high spectral brightness in 1060 nm α-DFB lasers with long resonators,” Electron. Lett. |

6. | J. Buus and M. Danielsen, “Carrier diffusion and higher-order transversal modes in spectral dynamics of semiconductor-laser,” IEEE J. Quantum Electron. |

7. | J.-M. Verdiell and R. Frey, “A broad-area mode-coupling model for multiple-stripe semiconductor lasers,” IEEE J. Quantum Electron. |

8. | R. K. Jain and R. C. Lind, “Degenerate four wave mixing in semiconductor doped glasses,” J. Opt. Soc. Am. |

9. | G. P. Agrawal and N. K. Dutta, |

10. | H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Techn. J. |

**OCIS Codes**

(140.2020) Lasers and laser optics : Diode lasers

(140.3430) Lasers and laser optics : Laser theory

(140.5960) Lasers and laser optics : Semiconductor lasers

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 22, 2005

Revised Manuscript: April 15, 2005

Published: May 2, 2005

**Citation**

Paul Petersen, Eva Samsøe, Søren Jensen, and Peter Andersen, "Guiding of laser modes based on self-pumped four-wave mixing in a semiconductor amplifier," Opt. Express **13**, 3340-3347 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-9-3340

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### References

- H. Nakajima and R. Frey, �??Collinear nearly degenerate four-wave mixing in intracavity amplifying media,�?? IEEE J. Quantum Electron. QE-22, 1349-1354 (1986). [CrossRef]
- M. Lucente, J. G. Fujimoto, and G. M. Carter, �??Spatial and frequency dependence of four-wave mixing in broad-area diode lasers,�?? Appl. Phys. Lett. 53, 1897-1899 (1988). [CrossRef]
- P. Kürz, R. Nagar, and T. Mukai, �??Highly efficient phase conjugation using spatially nondegenerate four-wave mixing in a broad-area laser diode�??, Appl. Phys. Lett. 68, 1180-1182 (1996). [CrossRef]
- R. J. Lang, K. Dzurko, A. A. Hardy, S. Demard, A. Schoenfelder, and D. Welch, �??Theory of grating-confined broad-area lasers,�?? IEEE J. Quantum Electron. QE-34, 2196-2210 (1998). [CrossRef]
- K. Paschke, R. Günter, J. Fricke, F. Bugge, G. Erbert and G. Tränkle, �??High power and high spectral brightness in 1060 nm α-DFB lasers with long resonators,�?? Electron. Lett. 39, 369-370 (2003). [CrossRef]
- J. Buus and M. Danielsen, �??Carrier diffusion and higher-order transversal modes in spectral dynamics of semiconductor-laser,�?? IEEE J. Quantum Electron. QE-13, 669-674 (1977). [CrossRef]
- J.-M. Verdiell and R. Frey, �??A broad-area mode-coupling model for multiple-stripe semiconductor lasers,�?? IEEE J. Quantum Electron. QE-26, 270-279 (1990). [CrossRef]
- R. K. Jain and R. C. Lind, �??Degenerate four wave mixing in semiconductor doped glasses,�?? J. Opt. Soc. Am. 73, 647-653 (1983). [CrossRef]
- G. P. Agrawal and N. K. Dutta, Semiconductor Lasers (Van Nostrand Reinhold, 2nd ed., New York, 1993).
- H. Kogelnik, �??Coupled wave theory for thick hologram gratings,�?? Bell Syst. Techn. J. 48, 2909-2947 (1969).

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